Orbital Exponents Research Articles

Basis Set Orbital Relaxation in Atomic and Molecular Hydrogen Systems

Orbital relaxation (OR) amounts to variation of the orbital exponents in hydrogen molecules and ions relative to the exponents of the isolated atom; it is represented as the sum of the one- and two-center contributions depending on the effective atomic charge and on the presence of other atoms in the molecule. The procedure for isolating the contributions of the exponent includes treatment of the OR of hydrogen in a special set of neutral and charged atoms and molecules with certain multiplicities of their electronic states. Within the framework of the spin-unrestricted Hartree-Fock method, we found and discussed the optimal values of the exponents of the basis orbitals of hydrogen atoms and molecules using the minimal split valence-shell basis set, the basis set that includes the polarization function, and the expanded set of grouped natural orbitals. A simple energy model is suggested for OR. Expressions are derived for evaluating the exponents of the relaxed orbitals in hydrogen-containing systems.

Self‐consistent shielding in atoms and molecules

AbstractThe orbital exponents of trial wave functions for simple systems can be found from the potential energy terms alone. Shielding of the nuclear charge by one electron on another is determined by the relative values of the nuclear–electron attraction and the electron–electron repulsion. For two electrons in the same orbital, the shielding is divided equally. For different orbitals, only the inner electron shields the outer. The systems tested are first‐row atoms, using Slater orbitals. It appears that if this approach can be generalized, it may not be necessary to calculate kinetic energies in chemical systems, since they will be determined by the orbital exponents. This would be useful if trial wave functions were not available, but trial electron density functions were. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Hylleraas method for many‐electron atoms. I. The Hamiltonian

AbstractA general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates ri, rij, and the polar angles θi, and φi. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the 2P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of −24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy −24.498369 a.u. A 150‐term wave function expansion leads to an energy of −24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Determination of outer shape of molecular orbitals based on two-dimensional Penning ionization electron spectroscopy for N 2 and CO by He*2 3S

Classical trajectory calculations for collision energy dependence of partial Penning ionization cross-sections (CEDPICS) of He*(2 3S) + (N 2,CO) were performed to obtain molecular orbital (MO) functions based on the observed data; coefficients in linear combination of atomic orbitals and orbital exponents for a minimal atomic basis set (STO-6G) were optimized in order to give good agreement between observed and calculated CEDPICS. Although a crude basis set was assumed, obtained MOs derived by experimental CEDPICS were found to be very similar in the outer characteristics to theoretically obtained SCF-MOs in terms of much larger basis sets (6-311+G*).

Linear Approximation of Basis Set Orbital Exponents for Isoelectronic Series of Atoms

The orbital exponents of Slater type atomic orbitals (AOs) in isoelectronic series of atoms may be approximated by the linear dependence on the nuclear charge using a technique developed for optimization of AO basis sets in Hartree–Fock–Roothaan calculations. This approach yields the analytical Hartree–Fock wave functions for any ion in the isoelectronic atomic series without optimization of orbital exponents. The approximated linear equations for atomic orbital basis sets of B, C, O, and F in the ground state are presented as an example.

Fast and stable algorithm for analytical evaluation of two‐center overlap integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers

AbstractA unified algorithm is presented for analytical evaluation of two‐center overlap integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers. Two‐center overlap integrals are expressed as finite sum of Gaunt coefficients and auxiliary functions S(p, t). Special attention is paid to the efficient calculation of this auxiliary function by introducing analytic and recurrence relations. In order to test the accuracy of the formula for two‐center overlap integrals, we performed an extensive study in which quantum numbers, orbital exponents, and internuclear distances were varied over wide ranges. We found that the method presented here for two‐center overlap integrals is fully reliable for very high quantum numbers and the computation times are very low. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Evaluation of Two-Center Overlap Integrals Over Slater-Type Orbitals Using Fourier Transform Convolution Theorem

Analytical expressions are presented for two-center overlap integrals over Slater-type orbitals using Fourier transform convolution theorem. The efficiency of calculation of these expressions is compared with those of other methods and good rate of convergence and great numerical stability is obtained for wide range of quantum numbers, orbital exponents and internuclear distances.

Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr

We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coefficients and orbital exponents are fully optimized in atomic Hartree–Fock (HF) calculations. As opposed to other quadruple zeta basis sets, the basis set errors in atomic ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the number of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in atomic MP2 calculations, and for a few cases determined at the HF level for excited atomic states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and density functional theory (DFT) calculations, and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in molecular HF, DFT, and MP2 calculations for a sample of diatomic and small polyatomic molecules by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compounds containing third row as well as alkaline and earth alkaline metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calculations indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in critical cases where other basis sets may show deficiencies.

Empirically adjusted and consistent set of EHT valence orbital parameters for all elements of the periodic table

The Hartree–Fock–Slater model of atoms has been modified by using individual values of the exchange parameter, αex, for each atom. Each value of αex was adjusted to reproduce the empirical value of the first ionization energy of the atom considered. The expectation values, energies and radial functions for all elements of the periodic table have been evaluated on the basis of the Hartree–Fock–Slater model and individual exchange parameters. A consistent set of Slater type orbital single ζ valence atomic orbital exponents and energies for all elements of the periodic table, suitable for orbital interaction analysis, is presented. These exponents were calculated by fitting the ⟨r⟩STO moments to numerical empirically adjusted ⟨r⟩HFS results. Qualitatively, the new parameters compare well with Fitzpatrick and Murphy exponents and Mann numerical Hartree–Fock ⟨r⟩HF moments and energy values but contain some influence of correlation and relativistic phenomena.

Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater Type Orbitals in Nonlined-Up Coordinate Systems

Two-center nuclear attraction integrals over Slater type orbitals with integer and noninteger principal quantum numbers in nonlined up coordinate systems have been calculated by means of formulas in our previous work (T. Ozdogan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The computer results for integer case are in best agreement with the prior literature. On the other hand, the results for noninteger case are not compared with the literature due to the scarcity of the literature, but also compared with the limit of integer case and good agreements are obtained. The proposed algorithm for the calculation of two-center nuclear attraction integrals over Slater type orbitals with noninteger principal quantum numbers in nonlined-up coordinate systems permits to avoid the interpolation procedure used to overcome the difficulty introduced by the presence of noninteger principal quantum numbers. Finally, numerical aspects of the presented formulae are analyzed under wide range of quantum numbers, orbital exponents and internuclear distances.

Accurate relativistic Gaussian basis sets determined by the third-order Douglas–Kroll approximation with a finite-nucleus model

Highly accurate relativistic Gaussian basis sets with a finite-nucleus model are developed for the 103 elements from H (Z=1) to Lr (Z=103). The present GTO sets augment the relativistic basis sets with a point-charge model proposed in the first paper of this series. The relativistic third-order Douglas–Kroll approach is adopted in optimizing the orbital exponents of a basis set by minimizing the atomic self-consistent field (SCF) energy. The basis sets are designed to have equal quality and to be appropriate for the incorporation of relativistic effects. The performance of the present basis sets is tested by calculations on a prototypical molecule, gold dimer using SCF and the singles and doubles coupled-cluster model with perturbative triples [CCSD(T)]. Several spectroscopic constants are calculated for the ground state of Au2. At the basis set superposition error (BSSE) corrected CCSD(T) level, the deviation from experiment is ΔRe=0.018 Å, Δωe=−3 cm−1, and ΔDe=−0.17 eV. The finite-size nucleus effect makes Re, ωe, and De smaller by 0.004 Å, 1 cm−1, and 0.05 eV, respectively. The application shows that the present relativistic Gaussian-type orbitals (GTO) basis sets with a finite-nucleus model are accurate and reliable.

Hartree–Fock High-Precision Analytical Functions of Open-Shell Atoms

The algorithm of high-precision optimization of basis functions suggested previously for calculating the analytical Hartree–Fock orbitals of closed-shell atoms is generalized to open-shell systems described by the Roothaan method (1960). Expressions for the first (free gradient) and second (Hesse matrix) derivatives of the system's energy with respect to the nonlinear parameters (orbital exponents) of the basis functions are derived in terms of density matrices for the filled and open shells. An algorithm is proposed for high-precision optimization of the nonlinear parameters using these equations based on Murtagh–Sargent and Newton minimization procedures. To illustrate the application of this algorithm, we give optimization of the basis sets of Slater type functions for atoms from the second row, as well as for Al, Si, P, K, Sc, and Fe atoms. The analytical Hartree–Fock orbitals giving nearly Hartree–Fock energies are calculated with a high degree of accuracy.

Evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers

AbstractA general formula has been established for the expansion of the product of two normalized associated Legendre functions centered on the nuclei a and b. This formula has been utilized for the evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals (STOs) with integer and noninteger principal quantum numbers. The formulas given in this study for the evaluation of two‐center overlap and nuclear attraction integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, orbital exponents, and internuclear distances. © 2001 Wiley Periodicals, Inc. Int J Quantum Chem, 2001

Simultaneous determination of nuclear and electronic wave functions without Born–Oppenheimer approximation: Ab initio NO+MO/HF theory

AbstractWe develop a simultaneous determination method of nuclear and electronic wave functions without the Born–Oppenheimer approximation. We examine two expanding methods, namely, molecular orbital (MO)‐type and valence bond (VB)‐type expansions for a nuclear orbital, which is a one‐particle wave function of a nucleus. The VB‐type expansion is shown to be more accurate than the MO‐type one because of the local nature of the nuclei. We also investigate the basis function expansion of the nuclear orbital and propose a scheme to determine the orbital exponent for the nuclear basis function. Numerical calculations confirm the accuracy and feasibility of the present method. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

Accurate relativistic Gaussian basis sets for H through Lr determined by atomic self-consistent field calculations with the third-order Douglas–Kroll approximation

Highly accurate relativistic Gaussian basis sets are developed for the 103 elements from H(Z=1) to Lr (Z=103). Orbital exponents are optimized by minimizing the atomic self-consistent field (SCF) energy with the scalar relativistic third-order Douglas–Kroll approximation. The basis sets are designed to have equal quality and to be appropriate for the incorporation of relativistic effects. The basis set performance is tested by calculations on prototypical molecules, hydrides, and dimers of copper, silver, and gold using SCF, Møller–Plesset theory, and the singles and doubles coupled-cluster methods with and without perturbative triples [CCSD, CCSD(T)]. Spectroscopic constants and dissociation energies are reported for the ground state of each species. The effects of relativity, electron correlation, and the basis set superposition error (BSSE) are investigated. At the BSSE corrected CCSD(T) level, the mean absolute error relative to experiment in De for three dimers (hydrides) is 0.13 (0.09) eV; for Re the error is 0.024 (0.003) Å, and for ωe it is 2 (13) cm−1. These illustrative calculations confirm that the present basis sets fulfill their design objectives.

Contracted polarization functions for the atoms Ca, Ga-Kr, Sr, and In-Xe

Contracted Gaussian-type function sets are proposed for polarization functions of the atoms Ga–Kr and In–Xe. We also report polarization functions for Ca and Sr. A segmented contraction scheme is used for its compactness and computational efficiency. The contraction coefficients and orbital exponents are fully optimized to minimize the deviation from accurate atomic natural orbitals. The present polarization functions yield more than 99% of atomic correlation energies predicted by accurate natural orbitals of the same size.

ANTIFERROMAGNETIC PHASE OF WIGNER ELECTRON CRYSTAL

The ground state energies of the 3-dimensional antiferromagnetic Wigner crystal corresponding to sc, bcc, fcc, diamond and perovskite structures are computed and the structural stability is analyzed. It is found that the bcc structure has the lowest energy. The possibility of the antiferromagnetic phase of the Wigner crystal having cubic or spherical surface as the region of occupation is analyzed. The effect of correlation is suitably taken into account. This still lowers the ground state energy. The calculations are done for the whole range of the density parameter r s from 1 to 200. The low density region favorable for Wigner crystallization is found to be above r s =10. The structure dependent Wannier function is approximated by orthonormalized Gaussian and it is employed in the calculation. The orbital exponent of the Gaussian is used as a variational parameter. The calculations are also done for the non-uniform positive background case.

Adapted Gaussian basis sets for ions with N≤54

Adapted Gaussian basis sets are reported for the cations He+–Xe+ and anions H−–I− using the generator coordinate Hartree–Fock method of selecting orbital exponents. For all ions studied, our wavefunctions are an improvement over those of da Silva and Trsic using a universal Gaussian basis set. Beside this, our ground state total energy values do not differ from the corresponding numerical Hartree–Fock values by more than 4.562mhartree for the cations and anions.

X-ray photoelectron spectroscopic studies of the oxidation of aluminium by liquid water monitored in an anaerobic cell.

The oxidation of clean polycrystalline aluminium foil was performed in a previously described anaerobic cell and studied by core level and valence band X-ray photoelectron spectroscopy. Oxide free sample surfaces were exposed under inert atmosphere at room temperature to increasingly oxidative environments of water vapor and liquid water beginning with an oxide free metal foil each time. Minor oxidative environments were found to produce a film of boehmite with a thinner outer film of gibbsite. Harsher oxidation in liquid water resulted in an oxide film much thicker than that found on the as received sample which contained an inner composition of gamma alumina with an outer film of gibbsite. As expected, the oxide film observed for ultra high purity polycrystalline aluminiurm foil in the as received state was gibbsite. The valence band of the cleaned metal is discussed in terms of residual surface species. The evaluation and determination of these residual surface species is evaluated following an analysis of two different band structure calculations which use different d orbital exponents. It is found that while the density of states is almost unaltered by changes in the d exponent, the predicted spectrum is substantially changed. The work makes extensive use of valence band X-ray photoelectron spectroscopy.

Dipole moments and molecular electrostatic potentials from MSINDO

To achieve an improved description of solvation effects in the newly derived version MSINDO of the SINDO semiempirical formalism an appropriate parameterization of dipole moments and electrostatic potentials was undertaken. The mean error of the dipole moment could be reduced by an appropriate choice of orbital exponents. It is also shown that the approximation of the molecular electrostatic potential (MESP) in the asymptotic density model (ADM) implemented in MSINDO results in a superior description compared to the previous SINDO1 implementation. The accuracy is demonstrated for a selected number of small molecules with carbon, nitrogen and oxygen atoms.